Diffusion Maps — Spectral Manifold Learning

Diffusion maps build a Markov chain on data via the Gaussian kernel k(x,y)=exp(-‖x-y‖²/ε). Eigenvectors of the normalized graph Laplacian give coordinates that respect the intrinsic geometry of the data manifold, ignoring ambient dimension. After t diffusion steps, diffusion distance D_t(x,y)² = Σ λ_k^{2t}(φ_k(x)-φ_k(y))² captures heat flow on the manifold. Left: original data (3D projected). Right: diffusion map embedding in ψ₁-ψ₂ space, colored by manifold parameter.